Mathematical Problems in Engineering

Volume 2018 (2018), Article ID 8192710, 8 pages

https://doi.org/10.1155/2018/8192710

## Sufficient Condition for the Parallel Flow Problem of Electromagnetic Loop Networks

^{1}Department of Electrical Engineering and Automation, Harbin Institute of Technology, Harbin 150001, China^{2}Harbin Institute of Technology at Zhangjiakou, Hebei 075421, China

Correspondence should be addressed to Yi Yang

Received 20 August 2017; Revised 18 November 2017; Accepted 23 November 2017; Published 27 February 2018

Academic Editor: Konstantinos Karamanos

Copyright © 2018 Yi Yang and Zhizhong Guo. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

#### Abstract

Electromagnetic loop networks (EMLNs) are pervasive in power networks. Their major characteristic is parallel flow. EMLNs with substantial parallel flow are considered to have a parallel flow problem. There is currently a serious disagreement about whether EMLNs have a parallel flow problem, which has resulted in different configurations of national grids. Therefore, this paper proposes a general model of EMLNs and derives the parallel current function, which formulates parallel flow, from the network equations of both the high and low voltage sides of an EMLN. Accordingly, the high and low voltage sides of an EMLN are equivalent to two sets of parallel identical multi-transmission lines. Finally, this paper considers operating margins and derives the sufficient condition under which parallel flow can be ignored. The sufficient condition not only determines whether an EMLN has a parallel flow problem but also reveals simple approaches to visually diminishing parallel flow. If the EMLN satisfies the sufficient condition, parallel flow can be ignored; otherwise, the EMLN needs to operate in a restricted way or to adopt open loop planning.

#### 1. Introduction

An electromagnetic loop network (EMLN) is a loop configuration of transmission lines with different voltage classes in a power network. Many EMLNs exist if the power transmission network consists of lines with two major voltage classes. Sometimes, high voltage and long distance transmission lines are required to operate in a parallel configuration with a low voltage class network, which is actually an EMLN. In addition, EMLNs exist in distribution networks, such as the isolated substations loop network in [1–3] and certain MV mesh distribution network in [4]. Therefore, EMLN research has broad practical significance.

A major characteristic of EMLNs is that the power flow on the high voltage side influences the power flow on the low voltage side. This phenomenon is hereinafter referred to as the parallel flow phenomenon. Due to the parallel configuration between the high and low voltage side networks, the power flow on the high voltage side network generates additional flow, designated as transporting flow, through the low voltage side network [5]. The transporting flow is analogous to the loop flow in [6]. If the power flow on the high voltage side network is aggravated, the transporting flow increases and deteriorates the operating conditions of the low voltage side network; therefore, the transmission capacity of the high voltage side of an EMLN is limited [7]. Furthermore, if the load demand increases, the lines on the low voltage side of the EMLN may be unable to maintain sufficient margins because of parallel flow. An EMLN is considered to have a parallel flow problem if the safe operation of the EMLN has to account for parallel flow.

Some researchers believe that most EMLNs have a parallel flow problem. They ascribe severe parallel flow to the large difference between the capacities of the lines on the high and low voltage side networks. Moreover, if a line on the high voltage side network is cut off, a large amount power may flow into the lines on the low voltage side network, significantly deteriorating their operating conditions. Most blackouts in history are related to the EMLN configuration [8–14], so these researchers assert that EMLNs should be avoided if possible. For example, the state grid of China prefers to open the EMLN in a controlled and planned manner once the EMLN satisfies some open loop conditions [15]. For this purpose, many Chinese researchers have qualitatively described the potential hazards of EMLNs [16] and have developed substantial approaches for opening EMLNs [17].

In contrast, other researchers believe that the parallel flow of EMLNs does not constitute a problem. They think that if the power flow is sufficiently small, EMLN parallel flow does not threaten the operation of the power system; at a heavier power flow circumstance, although parallel flow exists, it is the peak load demand or another critical factor that pushes the power system towards critical conditions [18, 19]. A power network without an EMLN cannot handle heavy power flow conditions as well as one with an EMLN. Moreover, contingency analysis is applicable to power networks regardless of the presence on EMLNs [20–24]. Therefore, these researchers believe that it is not necessary to specifically extract and study EMLNs or to deliberately avoid them. For example, despite the many EMLNs in the power transmission network in Germany [25], satisfactory operating conditions are maintained. Moreover, the problems of EMLNs have not drawn much attention, at least in English-language academic journals.

Do EMLNs have a parallel flow problem? There is currently no sufficient rational and objective approach to answer this question. The researchers who hold affirmative opinions base their conclusions on equivalent models and the qualitative drawbacks of EMLNs [16, 26]; however, this approach does not prove that the parallel flow of an EMLN constitutes a problem. The researchers who hold negative opinions ascribe the outages associated with EMLNs to other factors [27, 28]. Meanwhile, they neglect the objective reality that parallel flow in an EMLN is likely to deteriorate the operating conditions of the low voltage side network.

This paper formulates the parallel flow of EMLNs and derives the sufficient condition for the EMLN parallel flow problem. First, according to the network equations of the high and low voltage side networks, the current components that are generated by the high voltage side network are extracted from the branches of low voltage side network. These components are expressed as a system of linear functions with respect to the branch currents on the high voltage side network. Second, the system of linear functions is utilized to make the high and low voltage sides of the EMLN equivalent to two sets of parallel identical multi-transmission lines. Third, according to the margins of the branches, the sufficient condition, with respect to the transmission lines, to prevent the EMLN parallel flow problem is induced. Finally, the proposed theory is used to determine whether parallel flow can be ignored in an actual EMLN case.

*Nomenclature*. To facilitate the description, let the th element of each incidence matrix be −1 if the positive direction of branch points to node or 1 if the branch points in the opposite direction. Unless specifically declared, the four main subscripts in the remainder of this paper are as follows.

Subscript | Implication | Example |

Nodal variable | , the nodal voltage vector | |

Branch variable | , the branch voltage vector | |

The high voltage side network | , the incidence matrix of the high voltage side network | |

The low voltage side network | , the incidence matrix of the low voltage side network |

In addition, we use subscripts and to divide a vector (or a matrix) into subvectors and submatrices. A subvector with subscript (subscript ) consists of elements that lie in the rows of the inner nodes (boundary nodes) of the vector (see the beginning of Section 2). A submatrix with double subscripts is formed in an analogous manner. Take , which represents the submatrix of the bus admittance matrix of the low voltage side network , as an example. The elements of lie in the rows of the inner nodes and columns of the boundary nodes in .

#### 2. Model for EMLN Parallel Flow

The basic characteristics of an EMLN are (1) a network on the high voltage side and a network on the low voltage side and (2) transformers as we model the transformer as a series connection with the internal impedance, which is converted into the low voltage side, and the ideal transformation ratio. A model of an EMLN is illustrated in Figure 1, where the blocks labeled “” and “” represent connected power networks with two voltage classes and arbitrary topologies. We assume that there are ideal transformers in the EMLN. The nodes on the high voltage (low voltage) side of the ideal transformer branches are hereinafter referred to as the boundary nodes of the high voltage (low voltage) side network. In addition, we designate the nodes that are not boundary nodes as inner nodes.